Fin Fenton Spectral

The CDF of a weighted sum of correlated lognormal random variables has lacked a tractable characterization since Fenton (1960). We show that eigenvalue conditioning of the correlation matrix, followed by Fourier-cosine inversion, yields an analytic, grid-free \(N\)-term spectral representation of that CDF: the Spectral Lognormal Distribution. The central point is not an elementary closed form, but a reusable analytic object whose dominant approximation error is controlled by residual correlation after conditioning rather than by the spectral inversion itself.

The total CDF error decomposes into six independent components. Within the target parameter regime studied in Section 5, five are saturated at double-precision noise; the remaining dominant term is the residual-correlation error induced by finite-\(K\) conditioning. This yields a smooth deterministic CDF and quantile map, addressing a constraint identified by Acerbi (2002, Section 5): non-parametric evaluation of the full class of spectral risk measures for weighted sums of correlated lognormals.

The approximation also converges in Wasserstein-1 distance, implying simultaneous convergence of all Lipschitz risk measures. A secondary structural extension beyond lognormal marginals is discussed separately and is not part of the paper's central claim.

The structural core — optimality, convergence, error decomposition, well-posedness, uniqueness, and all four Acerbi coherence axioms — is formally structure-verified in Lean 4; Appendix D states the scope of that verification and the boundary between machine-checked structure and classical analysis. A computational study appears in the companion paper (Nagy, 2026b).